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A Representation Theorem for Reasoning in First-Order Multi-Agent Knowledge Bases Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph

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A Representation Theorem for Reasoning in First-Order Multi-Agent Knowledge Bases

Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Christoph Schwering Maurice Pagnucco

UNSW Sydney, Australia

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A Representation Theorem for Reasoning in First-Order Multi-Agent Knowledge Bases Reasoning in multi-agent epistemic knowledge bases reduces to classical validity

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A Representation Theorem for Reasoning in First-Order Multi-Agent Knowledge Bases Reasoning in multi-agent epistemic knowledge bases reduces to classical validity

◮ Logical framework: Levesque’s logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing logic of only-knowing

KAα OAα

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A Representation Theorem for Reasoning in First-Order Multi-Agent Knowledge Bases Reasoning in multi-agent epistemic knowledge bases Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces Turing-reduces to classical validity

◮ Logical framework: Levesque’s logic of only-knowing

KAα OAα

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A Representation Theorem for Reasoning in First-Order Multi-Agent Knowledge Bases Reasoning in multi-agent epistemic knowledge bases Turing-reduces to classical validity

◮ Logical framework: Levesque’s logic of only-knowing

KAα OAα

◮ Could implement reasoning service with off-the-shelf

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A knows A A knows that

but A doesn’t know B

B knows B

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KB:

OA

A = 7 ∧ ∀x
B = x → OB B = x
A only knows that A = 7 and

that if B = x, then B only knows that B = x

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KB:

OA

A = 7 ∧ ∀x
B = x → OB B = x
entails

Query:

KA∃z

B = z ∧ ¬KA B = z ∧ KB B = z
de dicto

de re de re

A knows that some number is equal to B, but A doesn’t know what the number is,

and B does know it

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Reduction: Eliminate Modal Operators

KB:

OA

A = 7 ∧ ∀x
B = x → OB B = x
entails

Query:

KA∃z

B = z ∧ ¬KA B = z ∧ KB B = z
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Reduction: Eliminate Modal Operators

KB:

OA

A = 7 ∧ ∀x
B = x → OB B = x
entails

Query:

KA∃z

B = z ∧ ¬KA B = z ∧ KB B = z
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Reduction: Eliminate Modal Operators

KB:

OA

A = 7 ∧ ∀x
B = x → OB B = x
entails

Query:

KA∃z

B = z ∧ ¬KA B = z ∧ KB B = z
KB’:

OA

A = 7 ∧ ∀x
B = x → P(x)
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Reduction: Eliminate Modal Operators

KB:

OA

A = 7 ∧ ∀x
B = x → OB B = x
entails

Query:

KA∃z

B = z ∧ ¬KA B = z ∧ KB B = z
KB’:

OA

A = 7 ∧ ∀x
B = x → P(x)
entails

Query’:

KA∃z

B = z ∧ ¬KA B = z ∧ ∃x(P(x) ∧

)

“For which x, z does OB B = x entail KB B = z?”

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Reduction: Eliminate Modal Operators

KB:

OA

A = 7 ∧ ∀x
B = x → OB B = x
entails

Query:

KA∃z

B = z ∧ ¬KA B = z ∧ KB B = z
KB’:

OA

A = 7 ∧ ∀x
B = x → P(x)
entails

Query’:

KA∃z

B = z ∧ ¬KA B = z ∧ ∃x(P(x) ∧

)

“For which x, z is

B = x → B = z valid?”

Call validity oracle!

[Levesque ’84]

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Reduction: Eliminate Modal Operators

KB:

OA

A = 7 ∧ ∀x
B = x → OB B = x
entails

Query:

KA∃z

B = z ∧ ¬KA B = z ∧ KB B = z
KB’:

OA

A = 7 ∧ ∀x
B = x → P(x)
entails

Query’:

KA∃z

B = z ∧ ¬KA B = z ∧ ∃x(P(x) ∧ x = z)
“For which x, z is

B = x → B = z valid?”

Call validity oracle!

[Levesque ’84]

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Summary

Assumption: agents always only-know something about each other.

✗ OA

P → OBα
✓ OA
(P → OBα) ∧ (¬P → OBβ)
✓ OA
A = 7 ∧ ∀x
B = x → OB B = x
Then:

Reasoning in multi-agent epistemic knowledge bases Turing-reduces to classical validity

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Appendix

SLIDE 17

Multi-Agent Knowledge Bases

OAα = A only-knows α [Levesque ’84]

◮ A considers all

all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all models of α possible

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Multi-Agent Knowledge Bases

OAα = A only-knows α [Levesque ’84]

◮ A considers all

all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all models of α possible OAφ entails KAψ ⇐

⇒ φ → ψ is valid

◮ provided that φ, ψ are objective!

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Multi-Agent Knowledge Bases

OAα = A only-knows α [Levesque ’84]

◮ A considers all

all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all models of α possible OAφ entails KAψ ⇐

⇒ φ → ψ is valid

◮ provided that φ, ψ are objective!

OAα implies OAβ ⇐

⇒ α and β are equivalent

◮ A can only-know at most one formula

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Multi-Agent Knowledge Bases

OAα = A only-knows α [Levesque ’84]

◮ A considers all

all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all all models of α possible OAφ entails KAψ ⇐

⇒ φ → ψ is valid

◮ provided that φ, ψ are objective!

OAα implies OAβ ⇐

⇒ α and β are equivalent

◮ A can only-know at most one formula

OAα is a multi-agent KB

multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB multi-agent KB ⇐

⇒ every model of α satisfies some OBβ

✗ OA

P → OBα
✓ OA
(P → OBα) ∧ (¬P → OBβ)
✓ OA∀x
f = x → OBα(x)
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SLIDE 21

Reduction

Replace each OAα(

x) with a fresh atom Pα( x)

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SLIDE 22

Reduction

Replace each OAα(

x) with a fresh atom Pα( x)

Replace each KAγ(

z) with a disjunction of ∃ x

Pα(

x) ∧ “for which x, z is α( x) → γ( z) is valid?”

ver all OAα(

x) at the same modal nesting level

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SLIDE 23

Reduction

Replace each OAα(

x) with a fresh atom Pα( x)

Replace each KAγ(

z) with a disjunction of ∃ x

Pα(

x) ∧ “for which x, z is α( x) → γ( z) is valid?”

ver all OAα(

x) at the same modal nesting level

Axiomatise that Pα(

x), Pβ( y) introduced for OAα( x), OAβ( y) Pα( x) →

Pβ(

y) ↔ “for which x, y is α( x) → β( y) is valid?”

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SLIDE 24

Summary

Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB:

Based on Levesque’s logic of only-knowing Every model of a multi-agent KB must satisfy some OBβ Allows for incomplete knowledge about other agent’s knowledge

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SLIDE 25

Summary

Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB:

Based on Levesque’s logic of only-knowing Every model of a multi-agent KB must satisfy some OBβ Allows for incomplete knowledge about other agent’s knowledge

Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning:

Oracle for FOL validity Turing reduction: calls oracle many times Would need FO-K45 oracle if it weren’t for

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SLIDE 26

Summary

Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB Multi-agent KB:

Based on Levesque’s logic of only-knowing Every model of a multi-agent KB must satisfy some OBβ Allows for incomplete knowledge about other agent’s knowledge

Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning Reduction to classical reasoning:

Oracle for FOL validity Turing reduction: calls oracle many times Would need FO-K45 oracle if it weren’t for

Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options Implementation options:

FOL theorem prover (e.g., Vampire) Limited belief logic

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